Introduction

A lot of people are familiar with the correlation coefficient, but they don’t know what it actually does. It’s a measure of how well two variables are correlated, where “correlated” means that one variable tends to change in a way that correlates with how much the other variable changes. A positive correlation means that when one variable goes up or down, so does the other; negative correlations mean that if one goes up, then usually the other goes down (or vice versa). The unit-free version of this statistic has no units because it’s always calculated using data from a continuous scale, such as height and weight

The correlation coefficient is a measure of how well two variables are correlated; the closer it is to 1, the stronger the relationship between the two variables.

The correlation coefficient is a measure of how well two variables are correlated; the closer it is to 1, the stronger the relationship between the two variables.

The formula for calculating a correlation coefficient is:

• r = æ æ æ æ (x1 * y1) – (x2 * y2)/(n-1) * 100%

where x1 and y1 are your independent variables (the predictor), x2 and y2 are your dependent variables (the outcome), n is the sample size, and r represents your correlation coefficient.

To interpret this number, you’ll want to know what type of value it represents: If r = 0 or 1 then there’s no relationship between independent and dependent variables–they’re completely unrelated! On the other hand if r = -1 then there’s perfect negative correlation between those same two sets of data points; if x increases then y decreases–and vice versa as well! Here’s how we’d read these values in practice:

The correlation coefficient measures how well a line fits a scatter plot of data points.

The correlation coefficient measures how well a line fits a scatter plot of data points. It’s a way of quantifying the strength of the relationship between two variables, with values ranging from -1 to 1. A value of 0 means there is no linear relationship between them; any other number indicates how closely related they are–the closer it is to 1, the stronger their relationship.

A positive correlation means that as one variable increases (say, height), so does another (weight). You can think about this as “more height equals more weight.” On the flip side, negative correlations mean that as one variable increases (say, age), another decreases (years lived). Again using age as an example: If you live longer than average life expectancy by 5 years at age 50 then your odds of living past 80 increase significantly compared with those who die before reaching 50 years old.*

The unit-free version of the correlation coefficient has no units because it’s always calculated using data from a continuous scale, such as height and weight.

The unit-free version of the correlation coefficient has no units because it’s always calculated using data from a continuous scale, such as height and weight.

The correlation coefficient is a number between -1 and 1 that describes how closely two variables are related. A positive value means that when one variable increases in value (e.g., height), so does the other (e.g., weight). A negative value indicates an inverse relationship between two variables: When one goes up, the other goes down; if one goes left, then so does its partner on the right side of your equation. The closer your r-value is to zero (or -1), the weaker your linear relationship between these two variables will be considered by researchers–but remember that even weak correlations aren’t necessarily meaningless!

Because case studies can’t be generalized, they aren’t valid statistical proofs.

Case studies are often used to support a hypothesis, but they can’t be used as statistical proofs. A statistical proof is a type of logical argument that uses statistics to prove something. It’s important to understand the difference between these two methods of proving something–one uses data and the other doesn’t.

For example:

• If I want to prove that my dog is smarter than your cat (and therefore should be treated better), I could gather some information about dogs and cats and then come up with an equation based on their intelligence scores. The equation will tell us how much more intelligent each animal is than another animal with which it shares certain characteristics (like being furry). This would be a statistical proof because it uses numbers from multiple cases in order establish relationships between variables like “intelligence” and “furryness.”
• Alternatively, if all we had was one example of each type of pet–our own pets–it would not be possible for us only using this case study alone without additional evidence (i.e., other studies) or reasoning outside our own personal experience/opinions since just being around animals doesn’t mean we know everything there is about them yet!

Takeaway:

The correlation coefficient is a measure of how well two variables are correlated. The closer it is to 1, the stronger the relationship between the two variables.

The correlation coefficient measures how well a line fits a scatter plot of data points: if you draw a straight line through your scatter plot and extend it on both sides by an equal amount (thus making your scatterplot into an interval), then the height at which this new line intersects each point gives you an estimate of what y would be if x were zero (i.e., if we knew nothing about x).

In this article, we’ve reviewed the basics of correlation and regression analysis. We hope that you now have a better understanding of how these tools can be used to study relationships between variables in a dataset.